| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw histogram then perform other calculations |
| Difficulty | Moderate -0.8 This is a straightforward grouped data question requiring standard techniques: drawing a histogram with unequal class widths (requiring frequency density calculation), estimating the mean using midpoints, and finding quartile positions. All are routine S1 procedures with no problem-solving or novel insight required, making it easier than average A-level maths questions. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02b Histogram: area represents frequency2.02f Measures of average and spread |
| Number of typing errors | \(1 - 5\) | \(6 - 20\) | \(21 - 35\) | \(36 - 60\) | \(61 - 80\) |
| Frequency | 24 | 9 | 21 | 15 | 42 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) class widths 5, 15, 15, 25, 20 | M1 | Attempt at class widths |
| \(\text{fd} = \frac{24}{5}, \frac{9}{15}, \frac{21}{15}, \frac{15}{25}, \frac{42}{20}\) | B1 | Correct widths of bars, with or without halves, seen on diagram |
| \(= 4.8, 0.6, 1.4, 0.6, 2.1\) | ||
| M1 | Attempt at fd or scaled freq | |
| A1 | Correct heights seen on graph ft their fd | |
| B1 | 5 | Correct labels, scales and halves |
| errors | ||
| (ii) mean \(= \frac{(3 \times 24 + 13 \times 9 + 28 \times 21 + 48 \times 15 + 70.5 \times 42)}{111}\) | M1 | Using mid points |
| M1 | using (\(\sum\) their fx) / their 111 | |
| \(= 40.2\) errors | A1 | 3 |
| (iii) LQ in 6 – 20 | B1 | |
| UQ in 61 – 80 | B1 | |
| Least value of IQ range is 61 – 20 = 41 | B1♦ | 3 |
(i) class widths 5, 15, 15, 25, 20 | M1 | Attempt at class widths
$\text{fd} = \frac{24}{5}, \frac{9}{15}, \frac{21}{15}, \frac{15}{25}, \frac{42}{20}$ | B1 | Correct widths of bars, with or without halves, seen on diagram
$= 4.8, 0.6, 1.4, 0.6, 2.1$ | |
| M1 | Attempt at fd or scaled freq
| A1 | Correct heights seen on graph ft their fd
| B1 | 5 | Correct labels, scales and halves
errors | |
(ii) mean $= \frac{(3 \times 24 + 13 \times 9 + 28 \times 21 + 48 \times 15 + 70.5 \times 42)}{111}$ | M1 | Using mid points
| M1 | using ($\sum$ their fx) / their 111
$= 40.2$ errors | A1 | 3 | correct answer
(iii) LQ in 6 – 20 | B1 |
UQ in 61 – 80 | B1 |
Least value of IQ range is 61 – 20 = 41 | B1♦ | 3 | If any or both wrong quartile ranges if sensible7 A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarised in the table below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Number of typing errors & $1 - 5$ & $6 - 20$ & $21 - 35$ & $36 - 60$ & $61 - 80$ \\
\hline
Frequency & 24 & 9 & 21 & 15 & 42 \\
\hline
\end{tabular}
\end{center}
(i) Draw a histogram on graph paper to represent this information.\\
(ii) Calculate an estimate of the mean number of typing errors for these 111 people.\\
(iii) State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range.
\hfill \mbox{\textit{CAIE S1 2014 Q7 [11]}}